9,328 research outputs found
Feedback Control of Traveling Wave Solutions of the Complex Ginzburg Landau Equation
Through a linear stability analysis, we investigate the effectiveness of a
noninvasive feedback control scheme aimed at stabilizing traveling wave
solutions of the one-dimensional complex Ginzburg Landau equation (CGLE) in the
Benjamin-Feir unstable regime. The feedback control is a generalization of the
time-delay method of Pyragas, which was proposed by Lu, Yu and Harrison in the
setting of nonlinear optics. It involves both spatial shifts, by the wavelength
of the targeted traveling wave, and a time delay that coincides with the
temporal period of the traveling wave. We derive a single necessary and
sufficient stability criterion which determines whether a traveling wave is
stable to all perturbation wavenumbers. This criterion has the benefit that it
determines an optimal value for the time-delay feedback parameter. For various
coefficients in the CGLE we use this algebraic stability criterion to
numerically determine stable regions in the (K,rho) parameter plane, where rho
is the feedback parameter associated with the spatial translation and K is the
wavenumber of the traveling wave. We find that the combination of the two
feedbacks greatly enlarges the parameter regime where stabilization is
possible, and that the stability regions take the form of stability tongues in
the (K,rho)--plane. We discuss possible resonance mechanisms that could account
for the spacing with K of the stability tongues.Comment: 33 pages, 12 figure
Time-delayed feedback control of unstable periodic orbits near a subcritical Hopf bifurcation
We show that Pyragas delayed feedback control can stabilize an unstable
periodic orbit (UPO) that arises from a generic subcritical Hopf bifurcation of
a stable equilibrium in an n-dimensional dynamical system. This extends results
of Fiedler et al. [PRL 98, 114101 (2007)], who demonstrated that such feedback
control can stabilize the UPO associated with a two-dimensional subcritical
Hopf normal form. Pyragas feedback requires an appropriate choice of a feedback
gain matrix for stabilization, as well as knowledge of the period of the
targeted UPO. We apply feedback in the directions tangent to the
two-dimensional center manifold. We parameterize the feedback gain by a modulus
and a phase angle, and give explicit formulae for choosing these two parameters
given the period of the UPO in a neighborhood of the bifurcation point. We
show, first heuristically, and then rigorously by a center manifold reduction
for delay differential equations, that the stabilization mechanism involves a
highly degenerate Hopf bifurcation problem that is induced by the time-delayed
feedback. When the feedback gain modulus reaches a threshold for stabilization,
both of the genericity assumptions associated with a two-dimensional Hopf
bifurcation are violated: the eigenvalues of the linearized problem do not
cross the imaginary axis as the bifurcation parameter is varied, and the real
part of the cubic coefficient of the normal form vanishes. Our analysis of this
degenerate bifurcation problem reveals two qualitatively distinct cases when
unfolded in a two-parameter plane. In each case, Pyragas-type feedback
successfully stabilizes the branch of small-amplitude UPOs in a neighborhood of
the original bifurcation point, provided that the phase angle satisfies a
certain restriction.Comment: 35 pages, 19 figure
Multi-dimensional classical and quantum cosmology: Exact solutions, signature transition and stabilization
We study the classical and quantum cosmology of a -dimensional
spacetime minimally coupled to a scalar field and present exact solutions for
the resulting field equations for the case where the universe is spatially
flat. These solutions exhibit signature transition from a Euclidean to a
Lorentzian domain and lead to stabilization of the internal space, in contrast
to the solutions which do not undergo signature transition. The corresponding
quantum cosmology is described by the Wheeler-DeWitt equation which has exact
solutions in the mini-superspace, resulting in wavefunctions peaking around the
classical paths. Such solutions admit parametrizations corresponding to metric
solutions of the field equations that admit signature transition.Comment: 15 pages, two figures, to appear in JHE
Stability and flow fields structure for interfacial dynamics with interfacial mass flux
We analyze from a far field the evolution of an interface that separates
ideal incompressible fluids of different densities and has an interfacial mass
flux. We develop and apply the general matrix method to rigorously solve the
boundary value problem involving the governing equations in the fluid bulk and
the boundary conditions at the interface and at the outside boundaries of the
domain. We find the fundamental solutions for the linearized system of
equations, and analyze the interplay of interface stability with flow fields
structure, by directly linking rigorous mathematical attributes to physical
observables. New mechanisms are identified of the interface stabilization and
destabilization. We find that interfacial dynamics is stable when it conserves
the fluxes of mass, momentum and energy. The stabilization is due to inertial
effects causing small oscillations of the interface velocity. In the classic
Landau dynamics, the postulate of perfect constancy of the interface velocity
leads to the development of the Landau-Darrieus instability. This
destabilization is also associated with the imbalance of the perturbed energy
at the interface, in full consistency with the classic results. We identify
extreme sensitivity of the interface dynamics to the interfacial boundary
conditions, including formal properties of fundamental solutions and
qualitative and quantitative properties of the flow fields. This provides new
opportunities for studies, diagnostics, and control of multiphase flows in a
broad range of processes in nature and technology
From resolvent estimates to damped waves
In this paper we show how to obtain decay estimates for the damped wave
equation on a compact manifold without geometric control via knowledge of the
dynamics near the un-damped set. We show that if replacing the damping term
with a higher-order \emph{complex absorbing potential} gives an operator
enjoying polynomial resolvent bounds on the real axis, then the "resolvent"
associated to our damped problem enjoys bounds of the same order. It is known
that the necessary estimates with complex absorbing potential can also be
obtained via gluing from estimates for corresponding non-compact models.Comment: 16 pages, 1 figur
Models and Feedback Stabilization of Open Quantum Systems
At the quantum level, feedback-loops have to take into account measurement
back-action. We present here the structure of the Markovian models including
such back-action and sketch two stabilization methods: measurement-based
feedback where an open quantum system is stabilized by a classical controller;
coherent or autonomous feedback where a quantum system is stabilized by a
quantum controller with decoherence (reservoir engineering). We begin to
explain these models and methods for the photon box experiments realized in the
group of Serge Haroche (Nobel Prize 2012). We present then these models and
methods for general open quantum systems.Comment: Extended version of the paper attached to an invited conference for
the International Congress of Mathematicians in Seoul, August 13 - 21, 201
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